MATHEMATICAL INDUCTION

THE NATURAL NUMBERS are the counting
numbers: 1, 2, 3, 4, etc. Mathematical induction is a technique for proving a statement -- a theorem, or a formula -- that is asserted about every natural number.

By "every", or "all," natural numbers, we mean any one that we might possibly name.

For example,

1 + 2 + 3 + . . . + n = ½n(n + 1).

This asserts that the sum of consecutive numbers from 1 to n is given by the formula on the right. We want to prove that this will be true for n = 1, n = 2, n = 3, and so on. Now we can test the formula for any given number, say n = 3:

1 + 2 + 3 = ½· 3· 4 = 6

-- which is true. It is also true for n = 4:

1 + 2 + 3 + 4 = ½· 4· 5 = 10.

But how are we to prove this rule for every value of n?

The method of proof is the following. It is called the principle of mathematical induction.

If

1)

when a statement is true for a natural number n = k, then it will also be true for its successor, n = k + 1;

and

2)

the statement is true for n = 1;

then the statement will be true for every natural number n.

To prove a statement by induction, we must prove parts 1) and 2) above. For, when the statement is true for n = 1, then according to 1), it will also be true for 2. But that implies it will be true for 3; which implies it will be true for 4. And so on. It will be true for any natural number that we might name.

The hypothesis of Step 1) -- "The statement is true for n = k" -- is called the induction assumption, or the induction hypothesis. It is what we assume when we prove a theorem by induction.

Example 1.
Prove that the sum of the first n natural numbers is given by this formula:

1 + 2 + 3 + . . . + n

=

n(n + 1) 2

.

Proof. We will do Steps 1) and 2) above. First, we will assume that the formula is true for n = k; that is, we will assume:

1 + 2 + 3 + . . . + k

=

k(k + 1) 2

. (1)

This is the induction assumption. Assuming this, we must prove that the formula is true for its successor, n = k + 1. That is, we must show:

1 + 2 + 3 + . . . + (k + 1)

=

(k + 1)(k + 2) 2

. (2)

To do that, we will simply add the next term(k + 1) to both sides of the induction assumption, line (1):